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A bargaining approach to determine common weights in DEA

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Abstract

In this paper Data Envelopment Analysis (DEA) is used to assess the relative efficiency of a set of decision-making units (DMUs). Each DMU is evaluated on the basis of the ratio of its weighted output (i.e. virtual output) over its weighted input (i.e. virtual input). Conventional DEA models allow each DMU to select the weighting scheme which optimizes its own evaluation. However, this total flexibility has drawbacks and in certain contexts may not be desirable. In such cases, it may be more appropriate to consider a common set of weights to benchmark and rank the alternatives using a common platform. In this paper, bargaining theory is used to determine a common set of weights in DEA. The advantage of using a bargaining approach is that the weights emerge bottom-up as the result of an agreement between the DMUs, instead of using an exogenous criterion imposed from above. The novelty of the proposed approach is that we consider two players per DMU, one whose utility function corresponds to its virtual input, and another whose utility is the negative of the virtual input. Thus, each DMU wants to choose the output weights so as to maximize its virtual output, and the input weights so as to minimize its virtual input. In this way, all DMU try to appear under the best possible light but the input and output weights are common and agreed. Models based on the Nash and the Kalai–Smorodinsky solutions are formulated and an application to a supplier selection problem is presented.

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Notes

  1. \(\mathbb {R}\) (\(\mathbb {R}_+\), \(\mathbb {R}_{++}\)) denotes the set of all (non-negative, positive) real numbers. \(\mathbb {R}^n\)(\(\mathbb {R}^n_+\), \(\mathbb {R}^n_{++}\)) is the Cartesian product of n copies of \(\mathbb {R}\) (\(\mathbb {R}_+\), \(\mathbb {R}_{++}\)).

  2. For vector inequalities, \(s\geqq s'\) means that \(s_i\ge s'_i\) for all \(i=1,\ldots , n\); \(s\ge s'\) means that \(s\geqq s'\) and \(s\ne s'\); \(s>s'\) means that \(s_i> s'_i\) for all \(i=1,\ldots , k\).

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Acknowledgements

The research of the authors has been supported by the Spanish Ministry of Science, Innovation and Universities under Project PGC2018-095786-B-I00.

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Authors and Affiliations

  1. Department of Economics, Quantitative Methods and Economic History, Pablo de Olavide University, Carr. de Utrera Km. 1, 41013, Sevilla, Spain

    I. Contreras & M. A. Hinojosa

  2. Department of Industrial Management, University of Seville, Sevilla, Spain

    S. Lozano

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  1. I. Contreras
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  2. S. Lozano
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Correspondence to I. Contreras.

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Contreras, I., Lozano, S. & Hinojosa, M.A. A bargaining approach to determine common weights in DEA. Oper Res Int J 21, 2181–2201 (2021). https://doi.org/10.1007/s12351-019-00498-w

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