考虑具有相同的投入和产出的两阶段决策单元的固定成本分摊问题。将待分摊的固定成本作为一种新的投入，建立两阶段加性DEA模型。首先证明了各决策单元从自身角度出发可以找到至少一种分摊方案，使其自身整体和阶段效率都达到Pareto有效。然后证明了在公共的权重下，至少存在一种分摊方案可以使得各决策单元整体和阶段效率都达到Pareto有效。并给出了这样的有效成本分摊方案集。为了得到公平的成本分摊方案，定义了各DMU各阶段对分摊方案的满意度，最大化最小的满意度，可以得到最终的成本分摊方案。将问题拓展到两阶段存在共享投入的生产系统中，这些性质定理同样成立。数值算例部分将本文与Yu等（2016）的研究结果进行了对比，验证了本文方法的有效性，指出了本文方法的优越性。本文的方法综合考虑了效率和公平，得到的固定成本分摊方案公平合理。
In this paper, the fixed cost allocation problem among the decision making units (DMU) with two stages is considered. First, the fixed cost is treated as an additional input to DMUs, and an additive two-stage DEA model is established. Then, some important theorems are proven:1. each DMU could find at least one allocation plan to make its total and each stage Pareto efficient; 2. there is at least one allocation plan for all DMUs and their stages to be Pareto efficient under a set of common weight; 3. the efficient fixed cost allocation set is provided. These theorems are also founded when it is expanded to the two-stage system with shared inputs. In order to obtain an equitable allocation, the satisfaction degree of each DMU's stage is defined. Through maximizing the minimal satisfaction degree, the final allocation plan is gotten. The numerical example shows that our approach is feasible and valid. Since the efficiency and fairness is considered together, the fixed cost allocation obtained from our approach is fair and valid. Our approach can give managers supports when they face cost allocation problems.
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