In this paper, for evaluating the efficiency in a three-stage DEA structure we use the additive and the multiplicative cooperative models that comply with the cooperation paradigm in the organizations, where for improving efficiency of system, stages cooperate together. Since the overall efficiency from the cooperative models may not be unique and consequently the stages’ efficiencies, then we combine them with the Nash bargaining game approach that besides maximizing efficiency scores for stages and the whole system, provides a unique and fair efficiency decomposition. Second order programming relaxation of the proposed nonlinear models are given in contrast to the parametric linear models in the literature. Finally, the effectiveness of the proposed models are illustrated with two numerical examples.
<p style='text-indent:20px;'>A vital issue in many organizations is the fair allocation of fixed cost among its subsets. In this paper, using data envelopment analysis, first we study fixed cost allocation based on both additive and multiplicative efficiency decompositions in the cooperative context for a two-stage structure in the presence of exogenous inputs and outputs. A conic relaxation formulation of multiplicative decomposition is given. Then, fixed cost allocation based on the leader-follower paradigm are presented. In the sequel, for allocating a fair fixed cost between the stages, using the results of the leader-follower model, we present the nonlinear Nash bargaining game model that independent of the efficiency score of each unit, allocates fixed cost to the stages. The nonlinear model is reformulated as a second order cone program which is an imporvement over the parametric linear models in the literature. Finally, two examples are used to illustrate the proposed models and compare their results with the existing models.</p>
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