Allocating a fixed cost across the decision making units with two-stage network structures

A prominent issue in many organizations involves the fair allocation of a total fixed cost among a group of entities. This paper extends the traditional fixed cost allocation problem to situations where the decision making units (DMUs) have a two-stage network structure. To this end, this paper first uses the data envelopment analysis (DEA) methodology to determine the relative efficiency while taking the internal structure and possible allocated costs into account. It shows that each DMU can separately maximize its relative efficiency to one through determining a series of allocations and selecting a set of relative weights. Next, we demonstrate that there exists an efficient allocation set based on a set of common weights, using which all DMUs and their two sub-stages can be simultaneously efficient. However, there are alternative allocation plans in the efficient allocation set. According to this non-uniqueness problem, we further optimize the allocation plans by taking the size of operation units into account, such that the allocation result is proportional to current input usages and output productions from a size point of view. In addition, we suggest a min–max model and a feasible computation algorithm for it to generate the final allocation plan in a way that minimizes the deviation between the efficient allocations and size allocations. More importantly, by repeatedly minimizing the maximum deviation, our proposed method can guarantee a unique allocation plan for all DMUs and sub-stages. Finally, both a numerical example modified from previous literature and an empirical application of bank activities are used to demonstrate the efficacy and usefulness of the proposed approach.