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DEA production games with fuzzy output prices

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Abstract

In DEA production models the technology is assumed to be implicit in the input-output data given by a set of recorded observations. DEA production games assess the benefits to different firms of pooling their resources and sharing their technology. The crisp version of this type of problems has been studied in the literature and methods to obtain stable solutions have been proposed. However, no solution approach exists when there is uncertainty in the unit output prices, a situation that can clearly occur in practice. This paper extends DEA production games to the case of fuzzy unit output prices. In that scenario the total revenue is uncertain and therefore the corresponding allocation among the players is also necessarily uncertain. A core-like solution concept is introduced for these fuzzy games, the Preference Least Core. The computational burden of obtaining allocations of the fuzzy total profit reached through cooperation that belong to the preference least core is high. However, the results presented in the paper permit us to compute the fuzzy total revenue obtained by the grand coalition and a fuzzy allocation in the preference least core by solving a single linear programming model. The application of the proposed approach is illustrated with the analysis of two cooperative production situations originated by data sets from the literature.

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Notes

  1. A fuzzy number, which we denote by \(\widetilde{z}\), is a fuzzy set on the space of real numbers \(\text{ I }\!\text{ R }\), whose membership function \(\mu _{\widetilde{z}}:\mathbb {R}\rightarrow [0,1]\) satisfies (i) there is a real number z, such that \(\mu _{\widetilde{z}}(z)=1\), (ii) \(\mu _{\widetilde{z}}\) is upper semicontinuous, (iii) \(\mu _{\widetilde{z}}\) is quasi-concave and (iv) \(supp(\widetilde{z})\) is compact, where \(supp(\widetilde{z})\) denotes the support of \(\widetilde{z}\). We denote the set of all fuzzy numbers by \(\mathbb {N}(\text{ I }\!\text{ R })\).

  2. It suffices that for any \(\tilde{c}\in \mathbb {N}(\text{ I }\!\text{ R })^P\) and \(y, z\in \text{ I }\!\text{ R }^P\), such that \(y\ge z\), \(\tilde{c} y\succeq \tilde{c} z\) holds.

  3. The \(\alpha \)-cut of a fuzzy number, \(\tilde{z}\in \text{ I }\!\text{ N }(\text{ I }\!\text{ R })\), is the real closed interval \(\tilde{z}_{\alpha }=\{z\in \text{ I }\!\text{ R }\,|\,\mu _{\tilde{z}}(z)\ge \alpha \}\), where \(\mu _{\tilde{z}}\) is the membership function for \(\tilde{z}\) (for \(\alpha = 0\) we set \(\tilde{z}_{0}= cl\{z\in \text{ I }\!\text{ R }\,|\,\mu _{\tilde{z}}(z)> 0\}\), where cl denotes the closure of sets).

References

  • Aubin, J. P. (1981). Cooperative fuzzy games. Mathematics of Operations Research, 6(1), 1–13.

    Article  MathSciNet  Google Scholar 

  • Cooper, W. W., Seiford, L. M., & Tone, K. (2000). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software. Dordrecht: Kluwer Academic Publishers.

    MATH  Google Scholar 

  • Färe, R., & Zelenyuk, V. (2003). On aggregate Farrell efficiencies. European Journal of Operational Research, 146(3), 615–620.

    Article  MathSciNet  Google Scholar 

  • González, A., & Vila, M. A. (1991). A discrete method for studying indifference and order relations between fuzzy numbers. Information Sciences, 56, 245–258.

    Article  MathSciNet  Google Scholar 

  • González, A., & Vila, M. A. (1992). Dominance relation on fuzzy numbers. Information Sciences, 64, 1–16.

    Article  MathSciNet  Google Scholar 

  • Hatami-Marbini, A., Emrouznejad, A., & Tavana, M. (2011). A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making. European Journal of Operational Research, 214(3), 457–472.

    Article  MathSciNet  Google Scholar 

  • Hinojosa, M. A., Mármol, A. M., Monroy, L., & Fernández, F. R. (2013). A multi-objective approach to fuzzy linear production games. International Journal of Information Technology and Decision Making, 12(5), 927–943.

    Article  Google Scholar 

  • Jahanshahloo, G. R., Vieira Junior, H., Hosseinzadeh Lofti, F., & Akbarian, D. (2007). A new DEA ranking system based on changing the reference. European Journal of Operational Research, 181(2007), 331–337.

    Article  Google Scholar 

  • Lertworasirikul, S., Fang, S. C., Nuttle, H. L., & Joines, J. A. (2003). Fuzzy BCC model for data envelopment analysis. Fuzzy Optimization and Decision Making, 2(4), 337–358.

    Article  MathSciNet  Google Scholar 

  • Li, S., & Zhang, Q. (2009). A simplified expression of the Shapley function for fuzzy games. European Journal of Operational Research, 196(1), 234–245.

    Article  MathSciNet  Google Scholar 

  • Lozano, S. (2013). DEA production games. European Journal of Operational Research, 231, 405–413.

    Article  MathSciNet  Google Scholar 

  • Lozano, S., Hinojosa, M. A., & Mármol, A. M. (2015). Vector-valued DEA production games. OMEGA, 52, 92–100.

    Google Scholar 

  • Lozano, S. (2014). Computing fuzzy process efficiency in parallel systems. Fuzzy Optimization and Decision Making, 13(1), 73–89.

    Article  MathSciNet  Google Scholar 

  • Monroy, L., Hinojosa, M. A., Mármol, A. M., & Fernández, F. R. (2013). Set-valued cooperative games with fuzzy payoffs: The fuzzy assignment game. European Journal of Operational Research, 225(1), 85–90.

    Article  MathSciNet  Google Scholar 

  • Nishizaki, I., & Sakawa, M. (2001). Fuzzy and multiobjective games for conflict resolution. Heidelberg: Physica-Verlag.

    Book  Google Scholar 

  • Owen, G. (1995). Game theory. Cambridge: Academic Press.

    MATH  Google Scholar 

  • Ramík, J., & R̆ímánek, J. (1985). Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets and Systems, 16(2), 123–138.

    Article  MathSciNet  Google Scholar 

  • Welch, E., & Barnum, D. (2009). Joint environmental and cost efficiency analysis of electricity generation. Ecological Economics, 68, 2336–2343.

    Article  Google Scholar 

  • Wu, H. C. (2012). Proper cores and dominance cores of fuzzy games. Fuzzy Optimization and Decision Making, 11(1), 47–72.

    Article  MathSciNet  Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was carried out with the financial support of the Spanish Ministry of Economy and Competitiveness under project ECO2015-68856-P (MINECO/FEDER).

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Correspondence to M. A. Hinojosa.

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Hinojosa, M.A., Lozano, S. & Mármol, A.M. DEA production games with fuzzy output prices. Fuzzy Optim Decis Making 17, 401–419 (2018). https://doi.org/10.1007/s10700-017-9278-8

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