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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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 \(\mathbb {N}(\text{ I }\!\text{ R })\).
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.
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).
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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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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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DOI: https://doi.org/10.1007/s10700-017-9278-8